If I am on two sites with a browser, can one of those sites see what I am doing on the other site?
If all is well, then currently: no, not in the way you describe (simultaneous browser sessions).
But, more general given the title of your question, bugs and privacy leaks might always be exploited. Like today in some browsers, embedding links to the other sites might reveal if you visited those, by abusing an old "visited links" privacy leak. December 2010:
The 46 sites exploit a widely known vulnerability that currently exists in all production version browsers except of Apple's Safari, which earlier this year became the first major browser to insulate users against the threat. Google Chrome, which is based on the same Webkit engine, soon followed. Beta versions of Mozilla Firefox and Microsoft Internet Explorer also fix the problem, but production versions of those browsers are still wide open.